Student Name: Zun Wang Student ID: 915109847

Assignment 8: Gene Networks 2

This should be a knittable .Rmd file. Include this file and a .html when you turn in the assignment.

cities <- read.delim("../input/us_cities.txt", row.names=1) # be sure to change the path
cities
cities_mat <- cities < 1500 # leave original matrix intact
diag(cities_mat) <- 0 # we do not have to fly within each of cities :)
cities_mat 
##      BOS NY DC MIA CHI SEA SF LA DEN
## BOS    0  1  1   0   1   0  0  0   0
## NY     1  0  1   1   1   0  0  0   0
## DC     1  1  0   1   1   0  0  0   0
## MIA    0  1  1   0   1   0  0  0   0
## CHI    1  1  1   1   0   0  0  0   1
## SEA    0  0  0   0   0   0  1  1   1
## SF     0  0  0   0   0   1  0  1   1
## LA     0  0  0   0   0   1  1  0   1
## DEN    0  0  0   0   1   1  1  1   0

Exercise 1: Based on this 0 or 1 representation of our network, what city is the most highly connected? Hint: sum the values down a column OR across a row for each city

colSums(cities_mat)
## BOS  NY  DC MIA CHI SEA  SF  LA DEN 
##   3   4   4   3   5   3   3   3   4

#CHI, Chicago is the most highly connected.

Try extending the range to 2000 miles in the above code (place the adjacency matrix in an object cities_mat_2000. Does the highest connected city change? If so explain.

cities_mat_2000 <- cities < 2000 # leave original matrix intact
diag(cities_mat_2000) <- 0 # we do not have to fly within each of cities :)
cities_mat_2000
##      BOS NY DC MIA CHI SEA SF LA DEN
## BOS    0  1  1   1   1   0  0  0   1
## NY     1  0  1   1   1   0  0  0   1
## DC     1  1  0   1   1   0  0  0   1
## MIA    1  1  1   0   1   0  0  0   0
## CHI    1  1  1   1   0   0  0  0   1
## SEA    0  0  0   0   0   0  1  1   1
## SF     0  0  0   0   0   1  0  1   1
## LA     0  0  0   0   0   1  1  0   1
## DEN    1  1  1   0   1   1  1  1   0
colSums(cities_mat_2000)
## BOS  NY  DC MIA CHI SEA  SF  LA DEN 
##   5   5   5   4   5   3   3   3   7

#Yes, it changes. Now DEN, denvor is the most highly connected city.

Exercise 2:

library(igraph)
## 
## Attaching package: 'igraph'
## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum
## The following object is masked from 'package:base':
## 
##     union
cities_graph2 <- graph.adjacency(cities_mat_2000, mode = "undirected")
plot(cities_graph2)

What is the total number of nodes in the plot? What is the total number of edges in the plot?

sum(cities_mat_2000)/2
## [1] 20

#There are 9 nodes and 20 edges.

Exercise 3: Re-calculate the adjacency matrix with the cutoff value at 2300. Calculate the number of edges using the following code. What do you get?

cities_mat_2300 <- cities < 2300 # leave original matrix intact
diag(cities_mat_2300) <- 0 # we do not have to fly within each of cities :)
cities_mat_2300
##      BOS NY DC MIA CHI SEA SF LA DEN
## BOS    0  1  1   1   1   0  0  0   1
## NY     1  0  1   1   1   0  0  0   1
## DC     1  1  0   1   1   0  0  0   1
## MIA    1  1  1   0   1   0  0  0   1
## CHI    1  1  1   1   0   1  1  1   1
## SEA    0  0  0   0   1   0  1  1   1
## SF     0  0  0   0   1   1  0  1   1
## LA     0  0  0   0   1   1  1  0   1
## DEN    1  1  1   1   1   1  1  1   0
cities_graph3 <- graph.adjacency(cities_mat_2300, mode = "undirected")
plot(cities_graph3)

sum(cities_mat_2300)/2 # divide by 2 because the matrix has 2 values for each edge
## [1] 24

#There are 9 nodes and 24 edges.

Exercise 4:

library(tidyverse)
## ── Attaching packages ────────────────────────────────────────────────────────────── tidyverse 1.3.0 ──
## ✓ ggplot2 3.3.0     ✓ purrr   0.3.3
## ✓ tibble  2.1.3     ✓ dplyr   0.8.5
## ✓ tidyr   1.0.2     ✓ stringr 1.4.0
## ✓ readr   1.3.1     ✓ forcats 0.5.0
## ── Conflicts ───────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::as_data_frame() masks tibble::as_data_frame(), igraph::as_data_frame()
## x purrr::compose()       masks igraph::compose()
## x tidyr::crossing()      masks igraph::crossing()
## x dplyr::filter()        masks stats::filter()
## x dplyr::groups()        masks igraph::groups()
## x dplyr::lag()           masks stats::lag()
## x purrr::simplify()      masks igraph::simplify()
# make sure to change the path to where you downloaded this using wget

# make sure to change the path to where you downloaded this using wget
brass_voom_E <- read_csv("../input/voom_transform_brassica.csv")
## Parsed with column specification:
## cols(
##   .default = col_double(),
##   GeneID = col_character()
## )
## See spec(...) for full column specifications.
brass_voom_E <- brass_voom_E[,-c(38,42,46)] # remove questionable library columns
head(brass_voom_E)
brass_voom_E$variance <- apply(brass_voom_E[,-1],1,var)

gene_exp300 <- brass_voom_E %>% filter(rank(desc(variance)) <= 300) %>% select(-variance) %>% column_to_rownames("GeneID") %>% as.matrix()

head(gene_exp300[,1:6])
##           IMB211_DP_1_INTERNODE IMB211_DP_1_LEAF IMB211_DP_1_PETIOLE
## Bra000059           6.008547559        -1.842657            7.643936
## Bra000441          -0.293202654         4.392560            2.161185
## Bra000615           8.199095650        -3.427619            3.948964
## Bra000638           8.403764872        -3.427619            5.137168
## Bra000815          -0.003696037        -3.427619           -1.981773
## Bra001986          11.287212959        -1.105691            5.073509
##           IMB211_DP_1_SILIQUE IMB211_DP_2_INTERNODE IMB211_DP_2_LEAF
## Bra000059           -1.746994              7.620447        -3.597011
## Bra000441           -1.162031             -1.601140         3.184349
## Bra000615            7.491411              8.081854        -1.275083
## Bra000638            4.308289              8.591153        -3.597011
## Bra000815           10.022844             -1.601140        -2.012048
## Bra001986            9.161399             11.097781        -0.789656
gene_exp5 <- gene_exp300[1:5,]
gene_exp5_cor <- cor(t(gene_exp5))
gene_exp5_cor %>% round(3)
##           Bra000059 Bra000441 Bra000615 Bra000638 Bra000815
## Bra000059     1.000     0.366     0.293     0.555    -0.003
## Bra000441     0.366     1.000    -0.070    -0.040     0.060
## Bra000615     0.293    -0.070     1.000     0.856     0.694
## Bra000638     0.555    -0.040     0.856     1.000     0.489
## Bra000815    -0.003     0.060     0.694     0.489     1.000
diag(gene_exp5_cor) <- 0
gene_exp5_cor
##              Bra000059   Bra000441  Bra000615   Bra000638    Bra000815
## Bra000059  0.000000000  0.36617944  0.2925751  0.55544256 -0.002815041
## Bra000441  0.366179441  0.00000000 -0.0702966 -0.03995147  0.060377637
## Bra000615  0.292575083 -0.07029660  0.0000000  0.85623598  0.693678504
## Bra000638  0.555442556 -0.03995147  0.8562360  0.00000000  0.489006036
## Bra000815 -0.002815041  0.06037764  0.6936785  0.48900604  0.000000000
gene_exp5_rank <- apply(gene_exp5_cor,2,function(x) rank(-abs(x)))
gene_exp5_rank
##           Bra000059 Bra000441 Bra000615 Bra000638 Bra000815
## Bra000059         5         1         3         2         4
## Bra000441         2         5         4         4         3
## Bra000615         3         2         5         1         1
## Bra000638         1         4         1         5         2
## Bra000815         4         3         2         3         5

(A) Describe what is meant by the “1” in the [“Bra000059”, “Bra000441”] cell.

1 indicated highest level of correlation in this subset.

(B) Do [“Bra000059”, “Bra000441”] and [“Bra000441”, “Bra000059”] have different values? If so, why?

#They have different values because the ranking is columnwise comparison, without using geometric distance, so the ranking order can be different.

gene_exp5_MR <- sqrt(gene_exp5_rank * t(gene_exp5_rank))
diag(gene_exp5_cor) <- 0
gene_exp5_MR
##           Bra000059 Bra000441 Bra000615 Bra000638 Bra000815
## Bra000059  5.000000  1.414214  3.000000  1.414214  4.000000
## Bra000441  1.414214  5.000000  2.828427  4.000000  3.000000
## Bra000615  3.000000  2.828427  5.000000  1.000000  1.414214
## Bra000638  1.414214  4.000000  1.000000  5.000000  2.449490
## Bra000815  4.000000  3.000000  1.414214  2.449490  5.000000

(C) Do [“Bra000059”, “Bra000441”] and [“Bra000441”, “Bra000059”] have different values in the MR tables? Why or why not?

#They have the same value, because by computing pairwise mutual ranks using geometric mean, the columnwise difference is canceled.

Exercise 5: (A) _Create the adjacency matrix described above and place it in an object called genes_adj_MR2.

genes_adj_MR2<- gene_exp5_MR < 2 # leave original matrix intact
diag(genes_adj_MR2) <- 0 # we do not have to fly within each of cities :)
genes_adj_MR2
##           Bra000059 Bra000441 Bra000615 Bra000638 Bra000815
## Bra000059         0         1         0         1         0
## Bra000441         1         0         0         0         0
## Bra000615         0         0         0         1         1
## Bra000638         1         0         1         0         0
## Bra000815         0         0         1         0         0

(B) Which genes are connected to Bra000615?

#Bra000638 and Bra000815.

Exercise 6:

(A)
Working with the the full gene_exp300 matrix, create an adjacency matrix called genes_adj_MR4 for the genes use a cutoff of MR < = 4. Remember to set the diagonal of the adjacency matrix to 0.Create a second adjacency matrix genes_adj_MR10 using a cutoff of of MR < = 10.

gene_exp300_cor <- cor(t(gene_exp300))
diag(gene_exp300_cor) <- 0
gene_exp300_rank <- apply(gene_exp300_cor,2,function(x) rank(-abs(x)))
gene_exp300_MR <- sqrt(gene_exp300_rank * t(gene_exp300_rank))
diag(gene_exp300_cor) <- 0
genes_adj_MR4<- gene_exp300_MR <= 4 # leave original matrix intact
diag(genes_adj_MR4) <- 0 # we do not have to fly within each of cities :)
genes_adj_MR4
##                               Bra000059 Bra000441 Bra000615 Bra000638 Bra000815
## Bra000059                             0         0         0         0         0
## Bra000441                             0         0         0         0         0
## Bra000615                             0         0         0         0         0
## Bra000638                             0         0         0         0         0
## Bra000815                             0         0         0         0         0
## Bra001986                             0         0         1         1         0
## Bra003909                             0         0         0         0         0
## Bra004715                             0         0         0         0         0
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genes_adj_MR10<- gene_exp300_MR <= 10 # leave original matrix intact
diag(genes_adj_MR10) <- 0 # we do not have to fly within each of cities :)
genes_adj_MR10
##                               Bra000059 Bra000441 Bra000615 Bra000638 Bra000815
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(B) Now we can do some calculations. If our cutoff is MR4, how many edges do we have in our 300 node network? What if we increase our cutoff to MR10? hint: sum( )

#for MR4
sum(genes_adj_MR4)/2
## [1] 323
#for MR10
sum(genes_adj_MR10)/2
## [1] 978

Exercise 7: Use the following code to plot our networks using different thresholds for connectivity. What do the colors represent? What patterns do you see in the visualization of this data? You will need to click on the zoom button on the plot to be able to visualize this well.

gene_graphMR4 <- graph.adjacency(genes_adj_MR4, mode = "undirected") #convert adjacency to graph
comps <- clusters(gene_graphMR4)$membership                        #define gene cluster membership
colbar <- rainbow(max(comps)+1)                                   #define colors
V(gene_graphMR4)$color <- colbar[comps+1]                          #assign colors to nodes
plot(gene_graphMR4, layout = layout_with_fr, vertex.size = 4, vertex.label = NA, main="MR 4")

gene_graphMR10 <- graph.adjacency(genes_adj_MR10, mode = "undirected") #convert adjacency to graph
comps <- clusters(gene_graphMR10)$membership                        #define gene cluster membership
colbar <- rainbow(max(comps)+1)                                   #define colors
V(gene_graphMR10)$color <- colbar[comps+1]                          #assign colors to nodes
plot(gene_graphMR10, layout = layout_with_fr, vertex.size = 4, vertex.label = NA, main="MR 10")

#Different colors represent different groups of genes, same color indicates genes with high correlation. The plot of MR10 has less groups and show more edges due to higher cutoff.

Exercise 8: The functions graph.density() and average.path.length() compute the graph density and average path length (big surprise. Use these functions to determine which graph (MR4 or MR10) has the greater density and the greater average path length. Are the results what you expected?

graph.density(gene_graphMR4)
## [1] 0.007201784
graph.density(gene_graphMR10)
## [1] 0.02180602
average.path.length(gene_graphMR4)
## [1] 6.168872
average.path.length(gene_graphMR10)
## [1] 10.94431

#This is as expected. MR10 plot has higher density and longer average path length, since it has a higher cutoff, there are more edges in it, and thus the density is higher, and also due to higher cut off, the average correlation level is not as high as MR4 plot, so it is expected to have a longer average path length.

Exercise 9:

gene_graphMR10 <- graph.adjacency(genes_adj_MR10, mode = "undirected")
distMatrix <- shortest.paths(gene_graphMR10, v = V(gene_graphMR10), to = V(gene_graphMR10))
head(distMatrix)[,1:7]
##           Bra000059 Bra000441 Bra000615 Bra000638 Bra000815 Bra001986 Bra003909
## Bra000059         0         3         3         2         9         3        15
## Bra000441         3         0         5         4        11         5        17
## Bra000615         3         5         0         1         7         1        13
## Bra000638         2         4         1         0         7         1        13
## Bra000815         9        11         7         7         0         6         6
## Bra001986         3         5         1         1         6         0        12
pl <- get.shortest.paths(gene_graphMR10, 227, 43)$vpath[[1]] # pull paths between node 132 and 45

V(gene_graphMR10)[pl]$color <- "green"                 # define highlight node color
E(gene_graphMR10)$color <- "grey"                      # define default edge color
E(gene_graphMR10, path = pl)$color <- "blue"           # define highlight edge color
E(gene_graphMR10)$width <- 1                           # define default edge width
E(gene_graphMR10, path = pl)$width <- 10               # define highlght edge width
plot(gene_graphMR10, layout = layout_with_fr, vertex.size = 6, vertex.label = NA)

colnames(genes_adj_MR4)[which.max(colSums(genes_adj_MR4))]
## [1] "Bra035069"
sort(betweenness(gene_graphMR4), decreasing = TRUE) %>% head(1)
## Bra003271 
##  1585.998
gene_graphMR4 <- graph.adjacency(genes_adj_MR4, mode = "undirected")
V(gene_graphMR4)$size <- 6                               # define default node size
V(gene_graphMR4)["Bra000059"]$color <- "cornflowerblue"  # define highlight node color
V(gene_graphMR4)["Bra000059"]$size <- 10                 # define highlight node size

gene_graphMR4$layout <- layout_with_fr(gene_graphMR4, niter=500) 
plot(gene_graphMR4, vertex.label = NA)

A Find the genes with the highest degree centrality and betweenness centrality in the MR10 networks

colnames(genes_adj_MR10)[which.max(colSums(genes_adj_MR10))]
## [1] "Bra033316"
sort(betweenness(gene_graphMR10), decreasing = TRUE) %>% head(1)
## Bra029719 
##  17246.33

#Highest degree centrality: Bra033316 #Highest betweenness centrality: Bra029719

B Plot the MR4 network, highlighting the highest degree centrality node and the highest betweenness centrality node (using different colors)

gene_graphMR4 <- graph.adjacency(genes_adj_MR4, mode = "undirected")
V(gene_graphMR4)$size <- 6                               # define default node size
V(gene_graphMR4)["Bra035069"]$color <- "cornflowerblue"  # define highlight node color
V(gene_graphMR4)["Bra035069"]$size <- 10                 # define highlight node size
V(gene_graphMR4)["Bra003271"]$color <- "salmon"  # define highlight node color
V(gene_graphMR4)["Bra003271"]$size <- 10

gene_graphMR4$layout <- layout_with_fr(gene_graphMR4, niter=500) 
plot(gene_graphMR4, vertex.label = NA)

C Plot the MR10 network, highlighting the highest degree centrality node and the highest betweenness centrality node (using different colors)

gene_graphMR10 <- graph.adjacency(genes_adj_MR10, mode = "undirected")
V(gene_graphMR10)$size <- 6                               # define default node size
V(gene_graphMR10)["Bra033316"]$color <- "cornflowerblue"  # define highlight node color
V(gene_graphMR10)["Bra033316"]$size <- 10                 # define highlight node size
V(gene_graphMR10)["Bra029719"]$color <- "salmon"  # define highlight node color
V(gene_graphMR10)["Bra029719"]$size <- 10

gene_graphMR10$layout <- layout_with_fr(gene_graphMR10, niter=500) 
plot(gene_graphMR10, vertex.label = NA)

D Do the highlighted nodes fit your expectation of betweenness centrality and degree centrality? Which one do you think better represents a central node in each network graph? Explain.

#Not all of them meet my expectation. In MR4 plot, the betweenness centrality node is more like a central node, since the degree centrality node seem not to lie in the biggest crowd where there are a lot of edges. While in MR10 plot, the degree centrality node makes more sense, it lies in the center of a big crowd, but betweenness centrality node does not seem to show its importance.